Let $$ R $$ be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when $$ R $$ is revolved about the $$ x $$-axis. $$ y=3 x^{2} $$ and $$ y=36-x^{2} $$ The volume of the solid is (Type an exact answer.)

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To find the volume of the solid generated when the region $$ R $$ is revolved about the $$ x $$-axis, we can use the method of disks (or washers) for volumes of solids.

Given:
- The curves are $$ y=3x^{2} $$ and $$ y=36-x^{2} $$.

First, we need to find the intersection points of the two curves to determine the limits of integration.

To find the intersection points, we set the two equations equal to each other:
$$ 3x^{2} = 36 - x^{2} $$

Solving for $$ x $$, we get:
$$ 4x^{2} = 36 $$
$$ x^{2} = 9 $$
$$ x = \pm 3 $$

Therefore, the intersection points are at $$ x = -3 $$ and $$ x = 3 $$.

Now, we can use the method of disks to find the volume of the solid. The volume of the solid is given by the integral:
$$ V = \pi \int_{-3}^{3} (36 - x^{2})^{2} - (3x^{2})^{2} \, dx $$

Let's calculate the volume using the method of disks.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\pi \times \int_{-3}^{3} \left(\left(36-x^{2}\right)^{2}-\left(3x^{2}\right)^{2}\right) dx$$
- step1: Subtract the terms:
  $$\pi \times \int_{-3}^{3} \left(1296-72x^{2}-8x^{4}\right) dx$$
- step2: Evaluate the integral:
  $$\pi \times \frac{28512}{5}$$
- step3: Multiply:
  $$\frac{\pi \times 28512}{5}$$
- step4: Multiply:
  $$\frac{28512\pi }{5}$$
The volume of the solid generated when the region $$ R $$ is revolved about the $$ x $$-axis is $$ \frac{28512\pi}{5} $$ cubic units.
The volume of the solid is $$ \frac{28512\pi}{5} $$ cubic units.

Let be the region bounded by the following curves. Use the method of your choice to find the volume of the solid
generated when is revolved about the -axis.
and
The volume of the solid is
(Type an exact answer.)

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Let be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when is revolved about the -axis. and The volume of the solid is (Type an exact answer.)